Properties of layer-by-layer vector stochastic models of force fluctuations in granular materials

On the distribution of elastic forces in disordered structures and materials. I. Computer simulation

Randomly structured materials and structures develop distributed internal forces when subjected to external loadings. Using granular packings, and random honeycombs and open–cell foams as prototypic examples, computer simulations were carried out to elucidate the statistical distribution of the internal forces in randomly structured materials. It was found that the sharpness of the force distribution depends critically on the degree of randomness of the structure. In general, the force distributions in these exemplary systems are found to range from the Maxwell–Boltzmann form to a sharply peaked Gaussian form. The results presented here form the basis for the development of a statistical mechanics theory to be presented elsewhere as part II.

Granular contact force density of states and entropy in a modified Edwards ensemble

A method has been found to analyze Edwards' granular contact force probability functional for a special case. As a result, the granular contact force probability density functions are obtained from first principles for this case. The results are in excellent agreement with the experimental and simulation data. The derivation assumes Edwards' flat measure--a density of states (DOS) that is uniform within the metastable regions of phase space. The enabling assumption, supported by physical arguments and empirical evidence, is that correlating information is not significantly recursive through loops in the packing. Maximizing a state-counting entropy results in a transport equation that can be solved numerically. For the present this has been done using the "mean-structure approximation," projecting the DOS across all angular coordinates to more clearly identify its predominant nonuniformities. These features are (1) the grain factor Psi related to grain stability and strong correlation between the contact forces on the same grain and (2) the structure factor Upsilon related to Newton's third law and strong correlation between neighboring grains.

Mechanical analog of temperature for the description of force distribution in static granular packings

It is shown that in a stressed granular packing, the effect of the applied pressure and structural randomness on the contact force distribution can be described accurately by a variational principle of minimizing energy subject to the constraint of keeping entropy at a fixed value. The constraint on entropy may be regarded as a measure of the degree of retained disorderness in the system. This procedure leads to the introduction of a parameter known as the "mechanical temperature." Similar to the role of the conventional thermal temperature in a thermal system, the mechanical temperature can be viewed as a parameter controlling the mixity between energy minimization and entropy maximization in the equilibrium condition.

Force correlations in the q model for general q distributions

We study force correlations in the q model for granular media at infinite depth for general q distributions. We show that there are no two-point force correlations as long as q values at different sites are uncorrelated. However, higher-order correlations can persist, and if they do, they only decay with a power of the distance. Furthermore, we find the entire set of q distributions for which the force distribution factorizes. It includes distributions ranging from infinitely sharp to almost critical. Finally, we show that two-point force correlations do appear whenever there are correlations between q values at different sites in a layer; various cases are evaluated explicitly.

Granular drag on a discrete object: shape effects on jamming

We study the drag force on discrete objects with circular cross section moving slowly through a spherical granular medium. Variations in the geometry of the dragged object change the drag force only by a small fraction relative to shape effects in fluid drag. The drag force depends quadratically on the object's diameter as expected. We do observe, however, a deviation above the expected linear depth dependence, and the magnitude of the deviation is apparently controlled by geometrical factors.

Stick-slip fluctuations in granular drag

We study fluctuations in the drag force experienced by an object moving through a granular medium. The successive formation and collapse of jammed states give a stick-slip nature to the fluctuations which are periodic at small depths but become "stepped" at large depths, a transition that we interpret as a consequence of the long-range nature of the force chains and the finite size of our experiment. Another important finding is that the mean force and the fluctuations appear to be independent of the properties of the contact surface between the grains and the dragged object. These results imply that the drag force originates in the bulk properties of the granular sample.

Force distributions in three-dimensional granular assemblies: effects of packing order and interparticle friction

We present a systematic investigation of the distribution of normal forces at the boundaries of static packings of spheres. A method for the efficient construction of large hexagonal-close-packed crystals is introduced and used to study the effect of spatial ordering on the distribution of forces. Under uniaxial compression we find that the form for the probability distribution of normal forces between particles does not depend strongly on crystallinity or interparticle friction. In all cases the distribution decays exponentially at large forces and shows a plateau or possibly a small peak near the average force but does not tend to zero at small forces.

Force Distributions near Jamming and Glass Transitions

We calculate the distribution of interparticle normal forces P(F) near the glass and jamming transitions in model supercooled liquids and foams, respectively. P(F) develops a peak that appears near the glass or jamming transitions, whose height increases with decreasing temperature, decreasing shear stress and increasing packing density. A similar shape of P(F) was observed in experiments on static granular packings. We propose that the appearance of this peak signals the development of a yield stress. The sensitivity of the peak to temperature, shear stress, and density lends credence to the recently proposed generalized jamming phase diagram.

Evolution of force distribution in three-dimensional granular media

Based on the discrete element method, the nature of normal contact force distribution and the effect of microstructure (contact fabric) on stresses in granular media sheared under constant mean stress condition is analyzed. The particles are tested in a periodic cell, having a nearly monodispersed system of spherical particles ("hard" and "soft"). The granular systems were initially isotropically compressed to have different solid fractions in order to obtain "dense" and "loose" samples. To study the nature of the force distribution, the granular medium was considered as both (i) noncohesive and (ii) with low values of interface energy. For the granular systems considered here, the nature of force distribution is shown to be dependent on shear history. The amount of interface energy introduced in the granular system does not seem to change the nature of normal force distribution significantly. However, it improves the postpeak stability in agreement with previous research [C. Thornton, Geotechnique 50, 43 (2000)]. The simulation of systems subjected to quasistatic shearing, in general, reveals that in a hard system (both dense and loose), the normal contact force distribution (i) at "peak" shear strength is purely an exponential decay throughout the entire range of force scale that is used, and (ii) at "isotropic" and "steady" states, the contact normal force distribution is bimodal with forces greater than average decaying exponentially at both the states, while the forces less than average tend to be half-Gaussian at the "isotropic" state and a second-order polynomial function at the "steady" state. For the soft (dense) system, the normal contact force distribution at "peak" shear strength is bimodal with forces greater than average decaying exponentially while the forces less than average tend to be a second-order polynomial function. However, for the soft system at both "isotropic" and "steady" states, the contact normal force distribution is half-Gaussian throughout the entire range of force scale that is used. It has been pointed out that in a granular system undergoing slow shearing, the shear strength of the system seems to depend on the ability of the material to form strong fabric anisotropy of contacts carrying strong (greater than average) force.

Scalar model of inhomogeneous elastic and granular media

We investigate theoretically how the stress propagation characteristics of granular materials evolve as they are subjected to increasing pressures, comparing the results of a two-dimensional scalar lattice model to those of a molecular dynamics simulation of slightly polydisperse disks. We characterize the statistical properties of the forces using the force histogram and a two-point spatial correlation function of the forces. For the lattice model, in the granular limit the force histogram has an exponential tail at large forces, while in the elastic regime the force histogram is much narrower, and has a form that depends on the realization of disorder in the model. The behavior of the force histogram in the molecular dynamics simulations as the pressure is increased is very similar to that displayed by the lattice model. In contrast, the spatial correlations evolve qualitatively differently in the lattice model and in the molecular dynamics simulations. For the lattice model, in the granular limit there are no in-plane stress-stress correlations, whereas in the molecular dynamics simulation significant in-plane correlations persist to the lowest pressures studied.

Exact calculation of the spatiotemporal correlations in the takayasu model and in the q model of force fluctuations in bead packs

We calculate exactly the two point mass-mass correlations in arbitrary spatial dimensions in the aggregation model of Takayasu. In this model, masses diffuse on a lattice, coalesce upon contact, and adsorb unit mass from outside at a constant rate. Our exact calculation of the variance of mass at a given site proves explicitly, without making any assumption of scaling, that the upper critical dimension of the model is 2. We also extend our method to calculate the spatiotemporal correlations in a generalized class of models with aggregation, fragmentation, and injection, which include, in particular, the q model of force fluctuations in bead packs. We present explicit expressions for the spatiotemporal force-force correlation function in the q model. These can be used to test the applicability of the q model in experiments.

Jamming and fluctuations in granular drag

We investigate the dynamic evolution of jamming in granular media through fluctuations in the granular drag force. The successive collapse and formation of jammed states give a stick-slip nature to the fluctuations which is independent of the contact surface between the grains and the dragged object, thus implying that the stress-induced collapse is nucleated in the bulk of the granular sample. We also find that while the fluctuations are periodic at small depths, they become "stepped" at large depths, a transition which we interpret as a consequence of the long-range nature of the force chains.